Articles about the pendulum in four journals devoted to the teaching of physics and one general science teaching journal (along with other miscellaneous articles from other journals) are listed in three broad categories — types of pendulums, the contexts in which these pendulums are used in physics teaching at secondary or tertiary levels and a miscellaneous category. A brief description of the sub-categories used is provided.

Pendulums in The Physics Education Literature: A Bibliography

We study the nonlinear oscillations of a forced and weakly dissipative spring–magnet system moving in the magnetic fields of two fixed coaxial, hollow induction coils. As the first coil is excited with a dc current, both a linear and a cubic magnet-position dependent force appear on the magnet–spring system. The second coil, located below the first, excited with an ac current, provides the oscillating magnetic driving force on the system. From the magnet–coil interactions, we obtain, analytically, the nonlinear motion equation of the system, found to be a forced and damped cubic Duffing oscillator moving in a quartic potential. The relative strengths of the coefficients of the motion equation can be easily set by varying the coils’ dc and ac currents. We demonstrate, theoretically and experimentally, the nonlinear behaviour of this oscillator, including its oscillation modes and nonlinear resonances, the fold-over effect, the hysteresis and amplitude jumps, and its chaotic behaviour. It is an oscillating system suitable for teaching an advanced experiment in nonlinear dynamics both at senior undergraduate and graduate levels.

https://iopscience.iop.org/article/10.1088/0143-0807/33/6/1473/pdf

Nonlinear dynamics of a magnetically driven Duffing-type spring–magnet oscillator in the static magnetic field of a coil

In this paper, we present a simple experiment to introduce the nonlinear behaviour of oscillating systems in the undergraduate physics laboratory. The transverse oscillations of a spring allow reproduction of three totally different scenarios: linear oscillations, nonlinear oscillations reducible to linear for small displacements, and intrinsically nonlinear oscillations. The chosen approach consists of measuring the displacements using video photogrammetry and computing the velocities and the accelerations by means of a numerical differentiation algorithm. In this way, one can directly check the differential equation of the motion without having to integrate it, or perform an experimental study of the potential energy in each of the analysed scenarios. This experiment allows first year students to reflect on the consequences and the limits of the linearity assumption for small displacements that is so often made in technical studies.

https://iopscience.iop.org/article/10.1088/0143-0807/35/1/015005/pdf

Experimental analysis of nonlinear oscillations in the undergraduate physics laboratory

We consider the nonlinear oscillations of a simple spring–magnet system that oscillates in the magnetic field of an inductive coil excited with a dc current. Using the relations for the interaction of a coil and a magnet we obtain the motion equation of the system. The relative strengths of the terms of this equation can be adjusted easily by varying the coil excitation current. Both the elastic constant and the shape of the potential energy function of the system can therefore be modified by varying that current. It is shown and demonstrated that this system is a case of anharmonic oscillations in a double-well potential. This nonlinear oscillator can be easily assembled with commonly available laboratory components, and monitored with a digital oscilloscope. Its simplicity is to be compared with the setups of many other nonlinear oscillators recently described. This oscillator is ideal for an advanced undergraduate laboratory experiment or for project work.

https://iopscience.iop.org/article/10.1088/0143-0807/33/5/1259/pdf

Anharmonic oscillations of a spring–magnet system inside a magnetic coil

[EN] This thesis presents, develops and validates a new methodology based on video analysis for the lessons related to dynamics during the first Physics courses at Universities. Instead of applying traditional approaches based on comparing the measured trajectory, r(t), with such associated to a determined model, in this thesis it is proposed to analyze directly the relation among the variables which define the dynamics of the system, i.e. the relation among position, speed and acceleration. In this way, it is possible to compare the differential equation that regulates the dynamic of the movement, providing a different perspective on the problems. Among others, one major advantage of the proposed approach is to ease the explanation of some phenomena because it is only necessary to describe the equation of the dynamics but without the requirement of its integration. This is considerable simpler for the students and lets them analyze other interesting dynamic systems which are not used to be studied during the first courses at University. On the other hand, this approach presents some tools, like the diagrams in thephases space, which ease the graphic interpretation of system evolution. Finally, the equations of the dynamics are quite more sensitive to the deviations between the model and the reality of its integrals. Thus, the students can assess and validate some hypotheses like the linearity or the absence of friction. In order to apply this approach, a computer vision system is developed and adapted to the requirements of a teaching lab of physics, but with the same performance as a research lab. The main limitations of current teaching systems of video analysis are identified, and the main problems related to the configuration of the cameras, their calibration, the automatic digitalization and the accuracy in the movement acquisition are solved. A result of this development is an automatic computer vision system ofhigh precision and interoperability with both cameras of low cost and a more sophisticated performance. The video acquisition system and the movement analysis integrate some specific algorithms for smoothing and numeric derivation. These algorithms are used to calculate accurately speed and acceleration of a movement. The final system (providing a smoother capture of movements) is validated through some experiments. The accuracy in position measuring is about 0.1%, 0.5% for speed and 2% for acceleration, even though using web cameras of very low cost. Nowadays, there is not any teaching lab equipment able to measure continuously and simultaneously these cinematic magnitudes with so high accuracy while being so cheap. In order to show the multitude of possibilities that are offered by this new approach, five practices of dynamics using this video analysis system are designed, covering aspects like the cinematic of the point and the solid, the non linear oscillations, the systems with dynamic balance, the systems with variable mass or the dynamic of rigid solid. [ES] En esta tesis se desarrolla y valida un nuevo enfoque, basado en el videoanalisis, para la docencia de la dinamica en los primeros cursos universitarios de Fisica. En lugar de usar los planteamientos tradicionales, basados en contrastar la trayectoria medida, r(t), con la asociada a un modelo predeterminado, se analiza directamente la relacion entre las variables que definen la dinamica del sistema, esto es, la relacion entre posicion, velocidad y aceleracion. De esta manera se puede contrastar la ecuacion diferencial que regula la dinamica del movimiento, lo que ofrece una vision diferente de los problemas. Entre otras ventajas, este enfoque simplifica la explicacion de los fenomenos, ya que solo es necesario describir la ecuacion de la dinamica sin necesidad de integrarla. Esto resulta mucho mas simple para los alumnos y permite analizar, ademas, sistemas dinamicos mas interesantes que no se suelen estudiar en los primeros cursos. Por otra parte, en este planteamiento se introducen herramientas como los diagramas en el espacio de fases, que facilitan la interpretacion de la evolucion de un sistema de una manera grafica e intuitiva para los alumnos. Finalmente, las ecuaciones de la dinamica son mucho mas sensibles a las desviaciones entre el modelo y la realidad que sus integrales. De esta manera, los alumnos pueden comprobar la validez de determinadas hipotesis como la linealidad o la ausencia de rozamiento, aspecto fundamental cuando se analizan movimientos reales. Para poder aplicar este planteamiento, se ha desarrollado un sistema de vision por computador adaptado a las necesidades de un laboratorio docente de Fisica, pero con las prestaciones propias de uno de investigacion. Se han analizado las limitaciones de los actuales sistemas docentes de videoanalisis y se han resuelto los principales problemas relacionados con la configuracion de las camaras, su calibracion, la deteccion automatica del movil en imagen y la precision en la captura de movimientos. Como resultado de este desarrollo, se dispone de un sistema de vision automatico, muy preciso y compatible tanto con camaras de bajo coste como con las de altas prestaciones. El sistema de captura y analisis de movimientos se complementa con unos algoritmos de suavizado y derivacion numerica que permiten calcular las velocidades y aceleraciones de forma muy precisa. El sistema conjunto (captura de movimientos mas suavizado), ha sido validado en una serie de experimentos. Su precision es del orden del 0.1% en la medicion de posiciones, del 0.5% para las velocidades y del 2% para las aceleraciones, incluso trabajando con camaras web de muy bajo coste. Ningun equipo de laboratorio docente puede medir de forma continua y simultanea las magnitudes cinematicas con tanta precision y a tan bajo coste. Para ilustrar las posibilidades de este nuevo enfoque, se han disenado cinco practicas de dinamica mediante videoanalisis, cubriendo aspectos como la cinematica del punto y del solido, las oscilaciones no lineales, los sistemas con equilibrio dinamico, los sistemas de masa variable o la dinamica del solido rigido. [CAT] En aquesta tesi es desenvolupa i valida un nou enfocament, basat en el video-analisi, per a la docencia de la dinamica en els primers cursos universitaris de Fisica. En compte d'usar els plantejaments tradicionals, basats en contrastar el moviment mesurat, r(t), amb l'associat a un model predeterminat, s'analitza directament la relacio entre les variables que defineixen la dinamica del sistema, aco es, la relacio entre posicio, velocitat i acceleracio. D'aquesta manera es pot contrastar l'equacio diferencial que regula la dinamica del moviment, la qual cosa ofereix una visio diferent dels problemes.' Entre altres avantatges, aquest enfocament simplifica l'explicacio dels fenomens, ja que nomes es necessari descriure l'equacio de la dinamica sense necessitat d'integrar-la. Aco resulta molt mes simple per als alumnes i permet analitzar, a mes, sistemes dinamics mes interessants que no se solen estudiar en els primers cursos. D'altra banda, en aquest plantejament s'introdueixen ferramentes com els diagrames en l'espai de fases, que faciliten la interpretacio de l'evolucio d'un sistema d'una manera grafica. Finalment, les equacions de la dinamica son molt mes sensibles a les desviacions entre el model i la realitat que els seus integrals. D'aquesta manera, els alumnes poden comprovar la validesa de determinades hipotesis, com la linealitat o l'absencia de fregament. Per a poder aplicar aquest plantejament, s'ha desenvolupat un sistema de visio artificial adaptat a les necessitats d'un laboratori docent de Fisica, pero amb les prestacions propies d'un d'investigacio. S'han analitzat les limitacions dels actuals sistemes docents de video-analisi i s'han resolt els principals problemes relacionats amb la configuracio de les cameres, el seu calibratge, la digitalitzacio automatica i la precisio en la captura de moviments. Com resultat d'aquest desenvolupament, es disposa d'un sistema de visio automatic, molt precis i compatible tant amb cameres de baix cost com amb les d'altes prestacions. El sistema de captura i analisi de moviments es complementa amb uns algoritmes de suavitzat i derivacio numerica que permeten calcular les velocitats i acceleracions de forma molt precisa. El sistema conjunt (captura de moviments mes suavitzat), ha sigut validat en una serie d'experiments. La seua precisio es de l'orde del 0.1% en el mesurament de posicions, del 0.5% per a les velocitats i del 2% per a les acceleracions, inclus treballant amb cameres web de molt baix cost. Cap equip de laboratori docent pot mesurar de forma continua i simultania les magnituds cinematiques amb tanta precisio i a tan baix cost. Per a il?lustrar les possibilitats d'aquest nou enfocament, s'han dissenyat cinc practiques de dinamica mitjancant video-analisi, cobrint aspectes com la cinematica del punt i del solid, les oscil?lacions no lineals, els sistemes amb equilibri dinamic, els sistemes de massa variable o la dinamica del solid rigid.

Aplicación de la Visión por Computador a la docencia de Fisica

A didactic apparatus is described which can be used by students to study both linear and nonlinear oscillations. The transitions from linear to nonlinear oscillations are considered both theoretically and experimentally.

https://iopscience.iop.org/article/10.1088/0143-0807/7/2/003/pdf

Linear and nonlinear oscillations: an experimental for students

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Simple apparatus for obtaining parametric resonance

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Linear and nonlinear oscillations: an experimental for students

Simple apparatus for obtaining parametric resonance